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We know that two shapes are similar if their corresponding angles are equal, and their corresponding sides in the same proportion. The same is true for similar triangles.

To prove that two
triangles are similar, we have to show that **one** (not all) of the
following statements is true:

**1. The three sides are in the same proportion.**

**2. Two sides are in the same proportion, and their included angle is equal.**

**3. The three angles of the first triangle are equal to the three angles of the second triangle.**

We have only shown **two** angles in the above triangles. If we calculate the missing angles, we find that angle
is 55° and angle
is 40°.
Therefore, all corresponding angles are equal.

- Question
Are triangles

**PQR**and**SQT**similar? If so, which of the above statements applies?

- Answer
This question is much easier if we draw the triangles separately.

**QR**corresponds to**QS**(**QR**= 2**QS**) and**QP**corresponds to**QT**(**QP**= 2**QT**). Angle**Q**is the same in both triangles. Therefore, they are similar, because two sides are in the same proportion and their included angle is equal (statement 2).Notice that they are similar, even though they are 'mirror' images (of different sizes).

Now try this one.

- Question
Are triangles

**ABC**and**DBE**similar? If so, which of the above statements (1, 2 or 3) applies?

- Answer
**ABC**and**DBE**are**similar**, because all of the corresponding angles are equal (statement 3).If you found this difficult, remember that angle

**B**is common to both triangles. The rules of parallel lines state that(corresponding angles)

and that (corresponding angles).

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